An excellent essay. I agree with almost everything you wrote; and you put it very well. Would you mind if I cross posted it to Vic Stenger's AVOID-L mailing list. You can check out the list here:
http://www.colorado.edu/philosophy/vstenger/
Brent Meeker
1Z wrote:
> ---------------------------------------------------------------------------------------------------------------------
>
> Metaphysics of Mathematics
>
> * Introduction: Why Mathematics Works
> * Six approaches to the Philosophy of Mathematics
> *
> o Approach 1: Empiricism: The "maths is physics" theory.
> o Approach 2: Platonism: Objectivity and objects.
> o Approach 3: Psychologism.
> o Approach 4: Formalism: Mathematics as a game.
> o Approach 5: Constructivism: How Real Are The Real Numbers?
> o Approach 6: Quasi Empiricism
> * More on Platonism
> *
> o Penrose on Platonism
> o The Source of Mathematical Certainty
> o Platonism and Causation
> * More on Formalism
> *
> o Formalism and Godel
>
> Why Mathematics Works
>
> 1. Mathematics is the theoretical exploration of every possible kind
> of abstract structure
> 2. The world has some kind of structure; it is not chaos or
> featureless or void
> 3. Therefore, it is likely that at least some of the structures
> discovered by mathematicians are applicable to the world
>
> (It would be strange if everything mathematicans came up with was
> empirically applicable, and might tend one towards Platonism or
> Rationalism, but this is not the case. It might be possible for
> mathematicians working completely abstractly and theoretically to fail
> to include any empirically useful structures in the finite list of
> structures so far achieved, but maths is not divorced from empiricism
> and practicallity to that extent -- what I say in (1) is something of
> an idealisation).
>
> Six Approaches to the Philosophy of Mathematics
>
> * Approach 1: Empiricism: The "maths is physics" theory.
> * Approach 2: Platonism: Objectivity and objects.
> * Approach 3: Psychologism.
> * Approach 4: Formalism: Mathematics as a game.
> * Approach 5: Constructivism: How Real Are The Real Numbers?
> * Approach 6: Quasi Empiricism
>
> Approach 1: Empiricism: The "maths is physics" theory.
> Mathematical empiricism is undermined by developments since the 19th
> century, of forms of mathematics with no obvious physical application,
> such an non-Euclidean geometry. I should probably say no obvious
> application at the time since non-Euclidean geometery -- or curved
> space -- was utilised in Einstein's General Theory of Relativity
> subsequent to its development by mathematicians. So M.E. has two
> problems: the existence of mathematical structures with no (currently)
> obvious physical application, and the fact that the physical
> applicability of different areas of mathematics varies with time,
> depending on discoveries in physics. Todays mathematical game-playing
> may be tomorrow's hard reality. It is also undermined by mathematical
> method, the fact that maths is a chalk-and-blackboard (or just
> thinking) activity, not a laboratory activity.
>
> Approach 2: Platonism: Objectivity and objects.
> Both Platonism and Empiricism share the assumption that mathematical
> symbols refer to objects. (And some people feel they have to believe in
> Empiricism simply because Platonism is so unacceptable). Platonism gets
> its force from noting the robustness and fixity of mathematical truths,
> which are often described as "eternal". The reasoning seems to be that
> if the truth of a statement is fixed, it must be fixed by something
> external to itself. In other words, mathematical truths msut be
> discovered, because if they were made they could have be made
> differently, and so would not be fixed and eternal. But there is no
> reason to think that these two metaphors --"discovering" and "making"--
> are the only options. Perhaps the modus operandi of mathematics is
> unique; perhaps it combines the fixed objectivity of discovering a
> physical fact about the external world whilst being nonetheless an
> internal, non-empirical activity. The Platonic thesis seems more
> obvious than it should because of an ambiguity in the word "objective".
> Objective truths amy be truths about real-world objects. Objective
> truths may also be truths that do not depend on the whims or
> preferences of the speaker (unlike statements about the best movie of
> flavour of ice-cream). Statements that are objective in the first sense
> tend to be objective in the second sense, but that does not mean that
> all statements that are objective in the second sense need be objective
> in the first sense. They may fail to depend on individual whims and
> preferences without depending on anything external to the mind.
>
> Approach3: Formalism: Mathematics as a game.
> Both Platonism and Empriricism share the assumption that mathematical
> symbols refer to objects. An alternative to both is the theory that
> they do not refer at all: this theory is called formalism. For the
> formalist, mathematical truths are fixed by the rules of mathematics,
> not by external objects. But what fixes the rules of mathematics ?
> Formalism suggests that mathematics is a meaningless game, and the
> rules can be defined any way we like. Yet mathematicians in practice
> are careful about the selection of axioms, not arbitrary. So do the
> rules and axioms of mathematics mean anything or not ?
>
> The reader may or may not have noticed that I have been talking about
> mathematical symbols "referring" to things rather than "meaning"
> things. This eliptically refers to a distinction between two different
> kinds or shades of meaning made by Frege. "Reference" is the
> external-world object a symbol is "about". "Sense" is the kind of
> meaning a symbol has even if does not have a reference. Thus statements
> about unicorns or the bald King of France have Sense but not Reference.
> Thus it is possible for mathematical statments to have a sense, and
> therefore a meaning, beyond the formal rules and defintions, but
> stopping short of external objects (referents), whether physical or
> Platonic. This position retains the negative claim of Formalism, that
> mathematical symbols don't refer to objects, and thus avoids the
> pitfalls of both Platonism and Empiricism. Howeverm it allows that
> mathematical symbols can have meanings of an in-the-head kind and thus
> explains the non-arbitrary nature of the choice of axioms; they are not
> arbitrary because they must correspond to the mathematician's intuition
> -- her "sense" -- of what a real number or a set is.
>
> So far we have been assuming that the same answer must apply uniformly
> to all mathematical statmentents and symbols: they all refer or none
> do. There is a fourth option: divide and conquer -- some refer and
> others don't.
>
> Approach 4: Psychologism: Nubers are Concepts
> Psychologism is the view that mathematical propositions are "about"
> ideas. (As opposed to physical objects, or Platonic objects) The
> counterargument is that there are inifinities in maths, but we do not
> have infinite numbers of ideas.
>
> All of these theories assume mathematical statements must refer to
> something, and all can be undercut by the hypothesis that mathematical
> statements can get by purely on "sense" and do not need "reference" at
> all (the sense/reference distinction is Frege's). So the
> Counter-counter argument is: we can have an idea of infinity without
> having an infinite number if ideas.
>
> "A fourth and final version of the third strategy, developed by
> Balaguer (1995, 1998) (and see Linsky and Zalta (1995) for a related
> view), is based upon the adoption of a particular version of platonism
> that can be called plenitudinous platonism, or as Balaguer calls it,
> full-blooded platonism (FBP). FBP can be intuitively but sloppily
> expressed with the slogan, 'All possible mathematical objects
> exist'; more precisely, the view is that all the mathematical objects
> that possibly could exist actually do exist, or that there exist as
> many mathematical objects as there possibly could. Balaguer argues that
> if platonists endorse FBP, then they can solve the epistemological
> problem with their view without positing any sort of
> information-transferring contact between human beings and abstract
> objects. For since FBP says that all possible mathematical objects
> exist, it follows that if FBP is true, then every purely mathematical
> theory that could possibly be true (i.e., that's internally consistent)
> accurately describes some collection of actually existing mathematical
> objects. Thus, it follows from FBP that in order to attain knowledge of
> abstract mathematical objects, all we have to do is come up with an
> internally consistent purely mathematical theory (and know that it is
> consistent). But it seems clear that (i) we humans are capable of
> formulating internally consistent mathematical theories (and of knowing
> that they're internally consistent), and (ii) being able to do this
> does not require us to have any sort of information-transferring
> contact with the abstract objects that the theories in question are
> about. Thus, if this is right, then the epistemological problem with
> platonism has been solved."
>
> http://plato.stanford.edu/entries/platonism/#2
>
> Not really, because there is now no role for Platonic objects. They are
> not involved in the actual process of verifying the truth of
> mathematical claims. A daemon could wish Platonia and all its contents
> away, and nothing would change.
> Approach 5: Constructivism: How Real Are The Real Numbers?
>
> Mathematical Platonism is the view that mathematical objects exist in
> some sense (not necessarily the sense that physical objects exist),
> irrespective of our ability to display or prove them.
>
> Constructivism is an opposing view, according to which only objects
> which can be explicitly shown or proven exist.
>
> Some constructivists believe in constructivism as an end in itself.
> Others use it as a means to an end,namely the elimination of
> infinities.
>
> Real numbers (numbers which cannot be expressed in a finite number of
> digits) are a particular problem for both kinds of constructivist. For
> the first, because most real numbers cannot be shown. Some are well
> known, such as pi and e (the base of natural logarithms). These can be
> generated by an algorithm; a sort of mathematical machine, where if you
> keep turning the handle, it keeps churning out digits. Of course, real
> numbers are not a finite string of digits, so the process never
> finishes. Nonetheless, the fact that the algorithm itself is finite
> gives us a handle on the real number, an ability to grasp it. However,
> the real number line is dense, so between any of these well known real
> numbers are infinities of intransigent real numbers, whose digits form
> no pattern, and which therefore cannot be set out, either as a string
> of digits, or as an algorithm.
>
> Or, at least, so the Platonists would have it. What alternatives are
> available for the constructuvist ?
>
> * 1. All numbers are equally (ontologically) real.
> * 2. The accessible real numbers (the transcendental numbers) are
> (ontologically) real, and the other (mathematically) "real" numbers are
> not.
> * 3. No (mathematically) real number is (ontologically) real.
>
> 1. is the position constructivists are trying to get away from. 3 is
> hardly tenable, since some irrational numbers, such as pi, *can* be
> constructed. And (2) contradicts the idea of density -- it suggests the
> real number line has gaps. Moreover, something like the requirment of
> denstity can be asserted without making any strong ontological
> commitments towards the reality of the real number line.
> Approach 6: Quasi Empiricism
> A number of recent developments in mathematics, such as the increased
> use of computers to assist proof, and doubts about the correct choice
> of basic axioms, have given rise to a view called quasi-empiricism.
> This challenges the traditional idea of mathematical truth as eternal
> and discoverable apriori. According to quasi-empiricists the use of a
> computer to perform a proof is a form of experiment. But it remains the
> case that any mathematical problem that can in principle be solved by
> shutting you eye and thinking. Computers are used because mathematians
> do not have infinite mental resources; they are an aid. Contrast this
> with traditonal sciences like chemistry or biology, where real-world
> objects have to be studied, and would still have to be studied by
> super-scientitists with an IQ of a million. In genuinely emprical
> sciences, experimentation and observation are used to gain information.
> In mathematics the information of the solution to a problem is always
> latent in the starting-point, the basic axioms and the formulation of
> the problem. The process of thinking through a problem simply makes
> this latent information explicit. (I say simply, but of ocurse it is
> often very non-trivial). The use of a computer externalises this
> process. The computer may be outside the mathematician's head but all
> the information that comes out of it is information that went into it.
> Mathematics is in that sense still apriori.
>
> Having said that, the quasi-empricist still has some points about the
> modern style of mathematics. Axioms look less like eternal truths and
> more like hypotheses which are used for a while but may eventualy be
> discarded if they prove problematical, like the role of scientific
> hypotheses in Popper's philosophy.
>
> Thus mathematics has some of the look and feel of empirical science
> without being empirical in the most essential sense -- that of needing
> an input of information from outside the head."Quasi" indeed!
>
> Peter D Jones 5/10/05-26/1/06
> More on Platonism
> Penrose on Platonism
> "our individual minds are notoriously imprecise, unreliable and
> inconsistent in their judgements" TBD
>
> So when we do maths we are not using our minds, but something else? Or
> maybe we are just using our minds in disciplined way -- after all, the
> discipline of maths has to be painfully learnt. TBD
>
> "Does this not point to something outside ourselves with a reality that
> lies beyond what each individual can achieve?" TBD
>
> What is achieved by individuals is achieved by individuals. How can
> some transcendent realm speak through the mouth of an individual? TBD
>
> "Nevertheless, one might take the alternative view that the
> mathematical world has no independent existence, and consists of
> certain ideas which have been distilled from our various minds, and
> which have found to be totally trustworthy, and are 'agreed by all',
> for example, or 'agreed by those in their right mind's, or 'agreed by
> those who have a PhD in mathematics' (not much use in Plato's day), and
> who have a right to venture an authoratiative opinion".
>
> Which mathematical world? The world of mathematics so far proven and
> understood, or the world that is waiting to be explored? However few
> mathematicians there are, there is no contradiction that they carry the
> world of known mathematics around in their heads. The fewer there are,
> the less there is to carry round. Perhaps Penrose thinks they must
> carry the whole Platonic world around to act as a "standard", but
> mathematical proof doesn't work by direct inspection of Platonia, it
> works slowly and painfully by axioms and deductions.
>
> "There seems to be a danger of circularity here; for to judge whether
> someone is in his or her right mind requires some external standard"
>
> External to them, yes...but when we make such judgements, we ue finite
> and earthly resources. The idea that Platonia provides a
> once-and-for-all absolute standard is not of much use unless it can be
> explained how the standard is applied. The actual standard against
> which a proposed arguemnt is tested involves the community of
> mathematicians checking it with their flawed and finite minds. Platonia
> might provide a higher standard, but it is not one anyone has ever
> succeeded in employing.
>
> "What I mean by this existence is really just the objectivity of
> mathematical truth".
>
> And what does thatmean? Existence is existence, truth is truth. People
> who disbelieve in Platonism can still believe in the objectivity of
> mathematics.
> Platonism and Causation
> Maths has no causal effect on the mind in the sense of an event in time
> causing a subsequent one. Arguably, there are other forms of causation.
> Arguably the laws of the universe "cause" events in the sense that if
> they were different, events would be different. Can the same argument
> be applied to mathematical truth? The problem is that mathematical
> truth is logically necessary -- it could not have been different!
> More on Formalism
> Formalism and Godel
> 'The incompleteness results affect the philosophy of mathematics,
> particularly viewpoints like formalism, which uses formal logic to
> define its principles. One can paraphrase the first theorem as saying
> that "we can never find an all-encompassing axiomatic system which is
> able to prove all mathematical truths, but no falsehoods."'
>
> 'On the other hand, from a strict formalist perspective this paraphrase
> would be considered meaningless because it presupposes that
> mathematical "truth" and "falsehood" are well-defined in an absolute
> sense, rather than relative to each formal system.'
>
> "Minds and Machines"
>
> * "Is the Brain A Machine?"
> * The Chinese Room and Consciousness
> * The Chinese Room and Semantics
> * Artificial Intelligence and Computers
> * Computationalism
> * Is the computationalist claim trivial -- are all systems
> computers?
> * Algorithms, recordings and counterfactuals
> * Turing and Other Machines
> * Time and Causality in Physics and Computation
> * >Maudlin's Argument and Counterfactuals
>
> "Is the Brain A Machine?"
> John Searle thinks so . Is he right ? To give a typically philosophical
> answer, that depends on what you mean by 'machine'. If 'machine' means
> an artificial construct, then the answer is obviously 'no'. However.
> Searle also thinks the the body is a machine, by which he seems to mean
> that it has been understand in scientific terms, we can explain biology
> by in terms of to chemistry and chemistry in terms of physics. Is the
> brain a machine by this definition ? The problem is that the job of he
> brain is to implement a conscious mind, just as the job of the stomach
> is to digest. The problem is that although our 'mechanical'
> understanding of the stomach does allow us to understand digestion we
> do not, according to Searle himself, understand how the brain produces
> consciousness. He does think that the problem of consciousness is
> scientifically explicable, so yet another definition of 'machine' is
> needed, namely 'scientifically explained or scientifically explicable'
> -- with the brain being explicable rather than explained. The problem
> with this stretch-to-fit approach to the meaning of the word 'machine'
> is that every time the definition of brain is broadened, the claim is
> weakened, made less impactful.
>
> PDJ 03/02/03
> The Chinese Room
> According to the proponents of Artificial Intelligence, a system is
> intelligent if it can convince a human interlocutor that it is. This is
> the famous Turing Test. It focuses on external behaviour and is mute
> about how that behaviour is produced. A rival idea is that of the
> Chinese Room, due to John Searle. Searle places himself in the room,
> manually executing a computer algorithm that implements
> intelligent-seeming behaviour, in this case getting questions written
> in Chinese and mechanically producing answers, without himself
> understanding Chinese. He thereby focuses attention on how the
> supposedly intelligent behaviour is produced. Although Searle's
> original idea was aimed at semantics, my variation is going to focus on
> consciousness. Likewise, although Searle's original specification has
> him implementing complex rules, I am going to take it that the Chinese
> Room is implemented as a conceptually simple system, in line with the
> theorem of CS which has it that any computer can be emulated by a
> Turing Machine.
>
> If you think a Chinese Room implemented with a simplistic, "dumb"
> algorithm can still be conscious, you are probably a behaviourist; you
> only care about that external stimuli get translated into the
> appropriate responses, not how this happens, let alone what it feels
> like to the system in question.
>
> If you think this dumb Chinese Room is not conscious, but a smart one
> would be, you need to explain why. Any smart AI can be implemented as a
> dumb TM, so the more complex inner workings which supposedly implement
> consciousness. could be added or subtracted without making any
> detectable difference. This seems to add up to epiphenomenalism, the
> view that consciousness exists but is a bystander that doesn't cause
> anything
>
> If you think that complexity makes a difference, so dumb AI's are
> definitely not conscious, you are probably a (strong) emergentist: the
> complexity is of such a nature that it cannot be reduced to a dumb
> algorithm such as a Turing machine, and is therefore non-computable.
>
> (A couple of points: I am talking about strong, Broad-style emergentism
> here, according to which the emerging property cannot be usefully
> reduced to something simpler. There is also a weaker sense of
> 'emergence' in which what emerges is a mere pattern of behaviour, like
> the way the waves 'emerge' from the sea. This is not quite black-box
> behaviourism, since some attention is being paid to what goes on inside
> the head, inasmuch as the right high-level pattern needs to be
> produced, but it is closer to behaviourism than to strong, irreducible,
> Broad-style emergentism.
>
> Also: I casually remarked that mental behaviour 'may not be
> computable'. This will shock some AI proponents, for whom the
> Chuch-Turing thesis proves that everything is computable. More
> precisely, everything that is mathematically computable is computable
> by a relatively dumb computer, a Turing that something can be simulated
> doesn't mean the simulation has all the relevant propeties of the
> original: flight simulators don't take off. Thirdly the mathematical
> sense of 'computable' doesn't fit well with the idea of
> computer-simulating fundamental physics. A real number is said to be
> mathematically computable if the algorithm that churns it out keeps on
> churning out extra digits of accuracy..indefinitely. Since such a
> algorithm will never finish churning out a single real number physical
> value, it is difficult to see how it could simulate an entire universe.
> Yes, I am assuming the universe is fundamentally made of real numbers.
> If it is, for instance finite, fundamental physics might be more
> readily computable, but the computability of physics still depends very
> much on physics and not just on computer science). Peter D Jones 8/6/05
> The Chinese Room and Semantics
>
> The CR concludes that an abstract set of rules is insufficient for
> semantics.
>
> The objection goes: "But there must be some kind of information
> processing structure that implements meaning in our heads. Surely that
> could be turned into rules for the operator of the Chinese Room".
>
> The Circularity Objection: an abstract structure must be circular and
> therefore must fail to have any real semantics. (It is plausible that
> any given term can be given an abstract definition that doesn't depend
> on direct experience. It is much less plausible that every *term* can
> be defined that way. Such a system would be circular in the same way
> as: "present: gift" "gift: present" but on a larger scale. If this
> argument, the Circularity Objection is correct, the practice of giving
> abstract definitions, like "equine quadruped" only works because
> somewhere in the chain of definitions are words that have been defined
> directly; direct reference has been merely deferred, not avoided
> altogether.)
>
> The objection continues: "But the information processing structure in
> our heads has a concrete connection to the real world: so do AI's
> (although the CR's are minimal) (This is the Portability Assumption)".
>
> But they are not the *same* concrete connections. The portability of
> abstract rules is guaranteed by the fact that they are abstract. But
> concrete causal connections are non-abstract, and are prima-facie
> unlikely to be portable -- how can you explain colour to an alien whose
> senses do not include anything like vision.
>
> If the Circularity Objection is correct, an AI (particularly a robotic
> one) could be expected to have *some* semantics, but there is no reason
> it should have *human* semantics. As wittgenstein said: "if a lion
> could talk, we could not understand it".
>
> Peter D Jones 13/11/05
> Artificial Intelligence and Computers
>
> An AI is not necessarily a computer. Not everything is a computer or
> computer-emulable. It just needs to be artificial and intelligent! The
> extra ingredient a conscious system has need not be anything other than
> the physics (chemistry, biology) of its hardware -- there is no forced
> choice between ghosts and machines.
>
> A physical system can never be exactly emulated with different hardware
> -- the difference has to show up somewhere. It can be hidden by only
> dealing with a subset of a systems abilities relevant to the job in
> hand; a brass key can open a door as well as an iron key, but brass
> cannot be substituted for iron where magnetism is relevant. Physical
> differences can also be evaded by taking an abstract view of their
> functioning; two digital circuits might be considered equivalent at the
> "ones and zeros" level of description even though they physically work
> at different voltages.
>
> Thus computer-emulability is not a property of physical systems as
> such. Even if all physical laws are computable, that does not mean that
> any physical systems can be fully simulated. The reason is that the
> level of simulation matters. A simulated plane does not actually fly; a
> simulated game of chess really is chess. There seems to be a
> distinction between things like chess, which can survive being
> simulated at a higher level of abstraction, and planes, which can't.
> Moreover, it seems that chess-like things are in minority, and that
> they can be turned into an abstract programme and adequately simulated
> because they are already abstract.
>
> Consciousness. might depend on specific properties of hardware, of
> matter. This does not imply parochialism, the attitude that denies
> consciousness to poor Mr Data just because he is made out of silicon,
> not protoplasm. We know our own brains are conscious; most of us intuit
> that rocks and dumb Chinese Rooms are not; all other cases are
> debatable.
>
> Of course all current research in AI is based on computation in one way
> or another. If the Searlian idea that consciousness is rooted in
> physics, strongly emergent, and non-computable is correct, then current
> AI can only achieve consciousness accidentally. A Searlian research
> project would understand how brains generate consciousness in the first
> place -- the aptly-named Hard Problem -- before moving onto possible
> artificial reproductions, which would have to have the right kind of
> physics and internal causal activity -- although not necessarily the
> same kind as humans.
>
> Computationalism
> Computationalism is the claim that the human mind is essentially a
> computer. It can be picturesquely expressed in the "yes, doctor"
> hypothesis -- the idea that, faced with a terminal disease, you would
> consent to having your consciousness downloaded to a computer.
>
> There are two ambiguities in "computationalism" -- consciousness vs.
> cognition, process vs programme -- leading to a total of four possible
> meanings.
>
> Most people would not say "yes doctor" to a process that recorded their
> brain on a tape a left it in a filing cabinet. Yet, that is all you can
> get out of the timeless world of Plato's heaven (programme vs process).
>
> That intuition is, I think, rather stronge than the intuition that
> Maudlin's argument relies on: that consciousness supervenes only on
> brain activity, not on counterfactuals.
>
> But the other ambiguity in computationalism offers another way out. If
> only cognition supervenes on computational (and hence counterfactual)
> activity, then consciousness could supervene on non-counterfactual
> activity -- i.e they could both supervene on physical processes, but in
> different ways.
> Platonic computationalism -- are computers numbers?
> Any computer programme (in a particular computer) is a long sequence of
> 1's and 0's, and therefore, a long number. According to Platonism,
> numbers exist immaterially in "Plato's Heaven". If programmes are
> numbers, does that mean Plato's heaven is populated with computer
> programmes?
>
> The problem, as we shall see is the "in a a particular computer"
> clause.
>
> As Bruno Marchal states the claim in a more formal language:
>
> "Of course I can [identify programmes with numbers ]. This is a key
> point, and it is not obvious. But I can, and the main reason is Church
> Thesis (CT). Fix any universal machine, then, by CT, all partial
> computable function can be arranged in a recursively enumerable list
> F1, F2, F3, F4, F5, etc. "
>
> Of course you can count or enumerate machines or algorithms, i.e.
> attach unique numerical labels to them. The problem is in your "Fix any
> universal machine". Given a string of 1's and 0s wihouta universal
> machine, and you have no idea of which algorithm (non-universal
> machine) it is. Two things are only identical if they have all*their
> properties in common (Leibniz's law). But none of the propeties of the
> "machine" are detectable in the number itself.
>
> (You can also count the even numbers off against the odd numbers , but
> that hardly means that even numbers are identical to odd numbers!)
>
> "In computer science, a fixed universal machine plays the role of a
> coordinate system in geometry. That's all. With Church Thesis, we don't
> even have to name the particular universal machine, it could be a
> universal cellular automaton (like the game of life), or Python,
> Robinson Aritmetic, Matiyasevich Diophantine universal polynomial,
> Java, ... rational complex unitary matrices, universal recursive group
> or ring, billiard ball, whatever."
>
> Ye-e-es. But if all this is taking place in Platonia, the only thing it
> can be is a number. But that number can't be associated with a
> computaiton by another machine, or you get infinite regress.
> Is the computationalist claim trivial -- are all systems computers?
> TBD
> Computational counterfactuals, and the Computational-Platonic Argument
> for Immaterial Minds
> For one, there is the argument that: A computer programme is just a
> long number, a string of 1's and 0's.
> (All) numbers exist Platonically (according to Platonism)
> Therefore, all programmes exist Platonically.
>
> A mind is special kind of programme (According to computaionalism)
> All programmes exist Platonically (previous argument)
> Therefore, all possible minds exist Platonically
> Therefore, a physical universe is unnecessary -- our minds exist
> already in the Platonic realm
>
> The argument has a number of problems even allowing the assumptions of
> Platonism, and computationalism.
>
> A programme is not the same thing as a process.
>
> Computationalism refers to real, physical processes running on material
> computers. Proponents of the argument need to show that the causality
> and dynamism are inessential (that there is no relevant difference
> between process and programme) before you can have consciousness
> implemented Platonically.
>
> To exist Platonically is to exist eternally and necessarily. There is
> no time or change in Plato's heave. Therefore, to "gain entry", a
> computational mind will have to be translated from a running process
> into something static and acausal.
>
> One route is to replace the process with a programme. let's call this
> the Programme approach.. After all, the programme does specify all the
> possible counterfactual behaviour, and it is basically a string of 1's
> and 0's, and therefore a suitable occupant of Plato's heaven. But a
> specification of counterfactual behaviour is not actual counterfactual
> behaviour. The information is the same, but they are not the same
> thing.
>
> No-one would believe that a brain-scan, however detailed, is conscious,
> so not computationalist, however ardent, is required to believe that a
> progamme on a disk, gathering dust on a shelf, is sentient, however
> good a piece of AI code it may be!
>
> Another route is "record" the actual behaviour, under some
> circumstances of a process, into a stream of data (ultimately, a string
> of numbers, and therefore something already in Plato's heaven). Let's
> call this the Movie approach. This route loses the conditional
> structure, the counterfactuals that are vital to computer programmes
> and therefore to computationalism.
>
> Computer programmes contain conditional (if-then) statements. A given
> run of the programme will in general not explore every branch. yet the
> unexplored branches are part of the programme. A branch of an if-then
> statement that is not executed on a particular run of a programme will
> constitute a counterfactual, a situation that could have happened but
> didn't. Without counterfactuals you cannot tell which programme
> (algorithm) a process is implementing because two algorithms could have
> the same execution path but different unexecuted branches.
>
> Since a "recording" is not computation as such, the computationalist
> need not attribute mentality to it -- it need not have a mind of its
> own, any more than the characters in a movie.
>
> (Another way of looking at this is via the Turing Test; a mere
> recording would never pass a TT since it has no
> condiitonal/counterfactual behaviour and therfore cannot answer
> unexpected questions).
>
> A third approach is make a movie of all possible computational
> histories, and not just one. Let's call thsi the Many-Movie approach.
>
> In this case a computation would have to be associated with all related
> branches in order to bring all the counterfactuals (or rather
> conditionals) into a single computation.
>
> (IOW treating branches individually would fall back into the problems
> of the Movie approach)
>
> If a computation is associated with all branches, consciousness will
> also be according to computationalism. That will bring on a White
> Rabbit problem with a vengeance.
>
> However, it is not that computation cannot be associated with
> counterfactuals in single-universe theories -- in the form of
> unrealised possibilities, dispositions and so on. If consciousness
> supervenes on computation , then it supervenes on such counterfactuals
> too; this amounts to the response to Maudlin's argument in wihch the
> physicalist abandons the claim that consciousness supervenes on
> activity.
>
> Of ocurse, unactualised possibilities in a single universe are never
> going to lead to any White Rabbits!
> Turing and Other Machines
> Turing machines are the classical model of computation, but it is
> doubtful whether they are the best model for human (or other organic)
> intelligence. Turing machines take a fixed input, take as much time as
> necessary to calculate a result, and produce a perfect result (in some
> cases, they will carry on refining a result forever). Biological
> survival is all about coming up with good-enough answers to a tight
> timescale. Mistaking a shadow for a sabre-tooth tiger is a msitake, but
> it is more accpetable than standing stock still calculating the perfect
> interpretation of your visual information, only to ge eaten. This
> doesn't put natural cognition beyone the bounds of computation, but it
> does mean that the Turing Machine is not the ideal model. Biological
> systems are more like real time systems, which have to "keep up" with
> external events, at the expense of doing some things imprefectly.
> Time and Causality in Physics and Computation
> The sum total of all the positions of particles of matter specififies a
> (classical) physical state, but not how the state evolves. Thus it
> seems that the universe cannot be built out of 0-width (in temporal
> terms) slices alone. Physics needs to appeal to something else.
>
> There is one dualistic and two monistic solutions to this.
>
> The dualistic solution is that the universe consists (separately) of
> states+the laws of universe. It is like a computer, where the data
> (state) evolves according to the programme (laws).
>
> One of the monistic solutions is to put more information into states.
> Physics has an age old "cheat" of "instantaneous velocities". This
> gives more information about how the state will evolve. But the state
> is no longer 0-width, it is infinitessimal.
>
> Another example of states-without-laws is Julian Barbour's Platonia.
> Full Newtonian mechanics cannot be recovered from his "Machian"
> approach, but he thinks that what is lost (universes with overall
> rotation and movement) is no loss.
>
> The other dualistic solution is the opposite of the second:
> laws-without-states. For instance, Stephen Hawking's No Boundary
> Conditions proposal
>
> Maudlin's Argument and Counterfactuals
> We have already mentioned a parallel with computation. There is also
> relevance to Tim Maudlin's claim that computationalism is incompatible
> with physicalism. His argument hinges on serparating the activity of a
> comptuaitonal system from its causal dispositions. Consciousness, says
> Maudlin supervened on activity alone. Parts of an AI mechansim that are
> not triggered into activity can be disabled without changing
> consciousness. However, such disabling changes the comutation being
> performed, because programmes containt if-then statements only one
> branch of which can be executed at a time. The other branch is a
> "counterfactual", as situationthat could have happened but didn't.
> Nonetheless, these counterfactuals are part of the algorithm. If
> changing the algorithm doesn't change the conscious state (because it
> only supervenes on the active parts of the process, not the unrealised
> counterfactuals), consciousness does not supervene on computation.
>
> However, If causal dispositions are inextricably part of a physical
> state, you can't separate activity from counterfactuals. Maudlin's
> argument would then have to rely on disabling counterfactuals of a
> specifically computational sort.
>
> We earlier stated that the dualistic solution is like the separation
> between programme and data in a (conventional) computer programme.
> However, AI-type programmes are typified by the fact that there is not
> a barrier between code and programme -- AI software is self-modifying,
> so it is its own data. Just as it is not physically necessary that
> there is a clear distinction between states and laws (and thus a
> separability of phsycial counterfactuals), so it isn't necessarily the
> case that there is a clear distinciton between programme and data, and
> thus a separability of computational counterfactuals. PDJ 19/8/06
>
>
> How to Build a Universe
>
> 1. Mysticism I: Is anything knowable?
> 2. Mysticism II: The Unreality of Real Things
> 3. Mechanism: The Simulation Argument
> 4. Mechanism, Organism and Knowability
> 5. Mathematics, the Universe and Everything
> 1. Why Mathematics Works
> 2. Gods and Monsters
> 3. For and against Mathematical Platonism
> 4. For and Against Matter
> 5. The Case Against Mathematial Monism
> 6. Logic, possibility, and Necessity
>
> Mysticism: Is Anything Knowable?
> (Based on a Dialogue with Bill Snyder)
> "Thinking about existence is quite other than existence itself. There
> is no such thing as "concrete" thinking; thinking always deals in
> abstractions and, hence, cannot contain existence itself".
>
> Well, when you think about rocks, you don't have rocks in the head. But
> does that put you at a disadvantage?
>
> "Thinking of rocks and being a rock are very different activities, or
> more broadly thinking about existence and existence itself are very
> different activities".
>
> That is true, but it does not show there is anything missing from our
> thinking. If there is nothing it is like to be a rock. there is nothing
> to stop us grasping all there is to know about rock-ness from our
> non-rock persepective.
>
> Of course, we feel that we ourselves have inner thoughts and
> (especially) feelings that cannot be understood by others, and we
> assume others do, even bats, as in the famous paper by Thomas Nagel.
> (More here). But the claim that is being examined language and thought
> are necessarily limited when it comes to grasping the world -- not that
> they are possibly limited depending on the nature of the world.
>
> It is logically possible, if coutnerintuitive, that there something it
> is like to be a rock, or anything else , and if so, there is something
> we will never know about anything. This is the conclusion being sought,
> but it is far from a necessary truth, only a possibility.
>
> But if there is something it is like to be a rock,. that is a
> metaphysical fact: the point being that we cannot discover the limits
> of thought from the nature of thought alone.
>
> "As for being able to describe something (event or thing or whatever),
> being able to refer to it is irrelevant. Referring to the keyboard on
> which I am typing (I just did it) is not to describe it. Describing it
> would be to state its colors, shape, and things like that. But to do
> that we must use words which have no precise meaning".
>
> Words only need to be precise enough to pick one thing out form
> another. It is probably overkill to describe every last atom.(But
> again, that is as much about what the world is like as the limitations
> of langauge).
>
> "To put my point broadly, describing perceptual objects is never
> completely possible, but only approximative because we use abstract
> concepts in the making of our description (like words which refer to
> shape and color). When one comes to the objects of wholly abstract
> thought such as the "objects" referred to in advanced scientific
> theories, then one is living in a dream world if one thinks that one
> can describe them or that the theory in question describes them".
>
> That's a rather different issue. The objects of fundamental science
> tend to be quite simple. The problem is not one of information
> overload, is the lack of familir reference-points. (Reference *does*
> matter).
>
> "The language used to refer to the objects MAY invite or suggest a
> certain descriptive content, but it is more MISleading than helpful in
> doing so. Calling an electron a particle may suggest that it is a very
> tiny hard little grain of sand like thing. But as we have come to know
> that is a very misleading way of thinking about it".
>
> Yes, the real description is mathematical.
>
> "It can just as well be thought of as a field or a wave; when thought
> of as a field, it is a field which at times generates particle like
> behavior and at other times wavelike behavior and at other times
> neither".
>
> The verbal descriptions are unified mathematically.
>
> "If QM is correct, its maths describes electrons completely. There is
> no hinterlan of additonal detail -- electrons have no 'hair'".
>
> "Language necessarily uses abstractions even in the description of
> perceptual objects. When one is dealing with scientific language
> (certainly in the advanced physical sciences) one is dealing with only
> abstractions"
>
> In what sense of "abstraction"?
>
> * The sense that fails to capture what-it-is to-be-like a thing?
> * The sense of omitted detail?
> * Or the sense of abstruse mathematics?
>
> The salient point is whether the thing described has any kind of
> hinterland which is left undescribed by the language employed. That is
> almost always the case with ordinary-langauge descriptions of everyday
> objects containing billions and billions of atoms -- but we only know
> that to be the case because of the success of scientific descriptions!
> It could be argued by analogy that just as the atom-by-atom picture is
> left undescribed by the ordinarys term "rock" or "tree", so there might
> be a further hinterland behind the scientific explanation. But that is
> only a "maybe", not necessity, and as usual, metaphysics, the nature of
> the world, is involved.
>
> "when one attempts to connect those abstractions with features drawn
> from the perceptual world and thereby to "describe" the objects, one
> only bamboozles any possibility of undertanding what is being referred
> to".
>
> There a various kinds of potential problem depending on the language
> being used, and on what it is being used to describe. Lack of
> understanding or communication is certainly possible, but is not a
> foregone conclusion.
>
> "Therefore, as long as one operates wholly within the conceptual
> structure one has no chance in hell of getting hold of existence
> itself.
>
> I think this is case of being bewitched by language -- specifically
> being misled by a very unforced choice of metaphor. One is not
> literally encased by ones own beliefs. The whole point of a concept is
> to refer to things other than itself.
>
> What does "getting hold" mean? As we have seen, knowing things from a
> third-personse, second-hand, arm-length perspective does not by itself
> imply that one is at a cognitve disadvantage. It depends on the thing
> being understood.
>
> "There are people who appear to me to operate wholly within one
> conceptual structure or another and live caged within their own
> thoughts, confusing what they think the world to be with existence
> itself".
>
> They may, as individuals be representing things incorrectly (insofars
> as they can be represented correctly). Or they may suffer from a more
> genreral problem of being out of contact with reality. Of course, if
> the general problem obtains in a strong way -- if philosophical
> scepticism is true -- there is probably not much difference between
> individuals. Everybody is equally wrong and right. (This is the classic
> "error" argument against solipsism).
>
> "But existence itself is necessarily beyond any thinking or conceiving"
>
> There is no necessity to it at all. How knowable a thing is depends on
> what it is and how you intend to know it.
>
> Peter D Jones 15/10/2006
> Mysticism II: the Unreality of Real Things, and the 3-Level View of
> Reality
> Some forms of mysticism claim that everything is a mystery to us, that
> we do not really know anything. Others claim that the problem lies in
> things themselves, that they lack reality rather than knowability.
> There is presumably some sense in which horses exist and unicorns do
> not. Thus the claim about the unreality of "conventional objects" seems
> to imply that they fall short of some higher standard. Is it a purely
> theoretical standard, or does it imply that there is some higher,
> transcedent reality , beyond conventional or empirical objects?
>
> A commmon-sense thinker might well claim that in the absence of any
> demonstrable higher reality, conventional existence is existence
> simpliciter, and the objects we conventionally take to be real are
> real.
>
> A famous proponent of this view was the Buddhist philosopher Nagarjuna.
> His philosophy was critical in nature, enabling him to establish the
> "nothing exists" claim using their ready-made criteria for existence of
> the philosophers he was criticising.
> Mechanism, Organism and Knowability
> Sceptical arguments often confuse being unable to know anything with
> being unable to know everything. But is that really a confusion? It's
> (logically) possible that things could be so constituted that the
> nature of every thing is intintimately dependent on the nature of every
> other thing. This is illustrated by an old parable called the "net of
> indra".
>
> FAR AWAY IN THE HEAVENLY ABODE OF THE GREAT GOD INDRA, THERE IS A
> WONDERFUL NET WHICH HAS BEEN HUNG BY SOME CUNNING ARTIFICER IN SUCH A
> MANNER THAT IT STRETCHES OUT INDEFINITELY IN ALL DIRECTIONS. IN
> ACCORDANCE WITH THE EXTRAVAGANT TASTES OF DEITIES, THE ARTIFICER HAS
> HUNG A SINGLE GLITTERING JEWEL AT THE NET'S EVERY NODE, AND SINCE THE
> NET ITSELF IS INFINITE IN DIMENSION, THE JEWELS ARE INFINITE IN NUMBER.
> THERE HANG THE JEWELS, GLITTERING LIKE STARS OF THE FIRST MAGNITUDE, A
> WONDERFUL SIGHT TO BEHOLD. IF WE NOW ARBITRARILY SELECT ONE OF THESE
> JEWELS FOR INSPECTION AND LOOK CLOSELY AT IT, WE WILL DISCOVER THAT IN
> ITS POLISHED SURFACE THERE ARE REFLECTED ALL THE OTHER JEWELS IN THE
> NET, INFINITE IN NUMBER. NOT ONLY THAT, BUT EACH OF THE JEWELS
> REFLECTED IN THIS ONE JEWEL IS ALSO REFLECTING ALL THE OTHER JEWELS, SO
> THAT THE PROCESS OF REFLECTION IS INFINITE
>
> This metaphysic does validate what otherwise be a non-sequitur; we
> cannot know anything because we cannot know everything. Any finite
> belief-system must be fall short of ultimate truth because it is
> finite. But without the holistic metaphysics of the Net Of Indra, the
> inference is simply a mistake. This is another mystical belief that
> requires a metaphysical posit, not simply a consideration of the
> limitations of knowledge.
>
> Science uses a methodology or reductionism, of divide and conquer, of
> breaking things down into their simplest components. A universe that
> plays ball with this is a mechanical, rather than organic universe. For
> our purposes, the most pertinent aspect of mechanism is the fact that
> it involves linear chains of causation. A causes B , and B causes C. C
> is not directly related to A and does not carry traces of A within it.
> This is an advantage, because it reduces the number of things one has
> to understand in order to understand C. (In the Organic universe, that
> is of course everyhting). It is also the source of pretty well every
> variety of scepticism, since A could have been replaced by A' without
> changing C. If C is some effect worked on my sense-organs by an
> external cause, then C could be the same in a scenario where it was
> caused by A' rather than A. I could be a brain in a vat, I could be the
> dupe of a Descartes Evil Spirit, I could be in a computer simulation.
> All these scenarios are variations on the theme of my percepts, my
> sense-data, being causes mechanically by something other that I think
> is causing them. In the organic universe, they are impossible because I
> already contain the whole world within myself, so substitions are
> impossible without changing everything else.
>
> Both Mechanism and Organism, then, lead to paradoxes. Mechanism makes
> the world simple enough to understand (structurally, from a 3rd-person
> perspective) at the expense of making its "in itself" nature uncertain.
> Organism means that every being contains the nature of the whol eof
> reality within itself -- but at the expense of making the Whole
> incomprehensible to linear, 3rd-person, structural knowing.
> Is Reality real ? the Simulation Argument
> The Simulation Argument seeks to show that it is not just possible that
> we are living inside a simulation, but likely.
>
> 1 You cannot simulate a world of X complexity inside a world of X
> complexity.(quart-into-a-pint-pot-problem).
>
> 2 Therefore, if we are in a simulation the 'real' world outside the
> simulation is much more complex and quite possibly completely different
> to the simulated world.
>
> 3 In which case, we cannot make sound inferences from the world we are
> appear to be in to alleged real world in which the simulation is
> running
>
> 4 Therefore we cannot appeal to an argumentative apparatus of advanced
> races, simulations etc, since all those concepts are derived from the
> world as we see it -- which, by hypothesis is a mere simulation.
>
> 5 Therefore, the simulation argument pulls the metaphysical rug from
> under its epistemological feet.
>
> The counterargument does not show that we are not living in a
> simulation, but if we are , we have no way of knowing whether it is
> likely or not. Even if it seems likely that we will go on to create
> (sub) simulations, that does not mean we are living in a simulation
> that is likely for the same reasons, since our simulation might be rare
> and peculiar. In particular, it might have the peculiarity that
> sub-simulations are easy to create in it. For all we know our
> simulators had extreme difficulty in creating our universe. In this
> case, the fact that it is easy to create sub simulations within our
> (supposed) simulation, does not mean it is easy to creae simulations
> per se.
> Mathematics, the Universe and Everything
>
> 1. Why Mathematics Works
> 2. Gods and Monsters
> 3. For and Against Mathematical Platonism
> 4. The Case Against Matter
> 5. The Case Against Mathematial Monism
>
> Why Mathematics Works
>
> 1. Mathematics is the theoretical exploration of every possible kind
> of abstract structure
> 2. The world has some kind of structure; it is not chaos or
> featureless or void
> 3. Therefore, it is likely that at least some of the structures
> discovered by mathematicians are applicable to the world
>
> (It would be strange if everything mathematicans came up with was
> empirically applicable, and might tend one towards Platonism or
> Rationalism, but this is not the case. It might be possible for
> mathematicians working completely abstractly and theoretically to fail
> to include any empirically useful structures in the finite list of
> structures so far achieved, but maths is not divorced from empiricism
> and practicallity to that extent -- what I say in (1) is something of
> an idealisation).
>
> Peter D Jones 8/12/04
> Gods and Monsters
> In the meditations of Marcus Aurelius, a distinction is drawn between
> two basic attitudes to philosophy, the Gods and the Monsters. Monsters
> are down-to-earth and commonsensical, Gods are high-minded and
> abstract. >
> gods Monsters
> Epistemology Rationalism Empiricism
> Metaphysics Idealism Materialism
> Logic Necessity Contingency
> TBD
> The Mathematical Monistic Many-World Model Introduced
>
> 1. Single (physical) universe theories
> 2. Physical Multiverse theories.
> 3. Mathematical Multiverse Theories
>
> Platonism is the idea that mathematical objects have an independent
> existence of their own, in addition to the physical world. The
> Mathematical Monistic Many-World Model idea keeps the Platonic realm of
> numbers, but drops the physical world, claiming we are somehow "inside"
> the world of numbers. One alleged benefit of this view is that it
> removes the contingency, the "just so" quality of a universe consiting
> of specific physical laws. Another is the claim that the success of
> physics shows everything is mathematical, or at least has a
> mathematical description. Another is the claim that matter adds nothing
> explanatorily to maths.
>
> For and Against Mathematical Platonism
> The case for Mathematical Platonism needs to be made in the first
> place; if numbers do not exist at all, the universe, as an existing
> thing, cannot be a mathematical structure. (solipsists read: if numbers
> are not real, I cannot be mathematical structure).
>
> The claim that nothing exists but numbers is not just the claim that
> the world is the kind of mathematical structure described by physics,
> because that is only one of a multitude of mathematically possible
> structures. The nothing-but-numbers claim entails the claim that a
> multitude of mathematically consistent worlds exist which do not
> correspond to anything that is observed. To put it another way, physics
> is based on picking the mathematical structure that corresponds to
> observation, so that only a subsection of physics is mathematically
> useful. Thus the success of physics as a mathematical science does not
> prove the nothing-but-numbers claim.
>
> Theories of Everything (TOEs) seek to avoid explain away the apparent
> (contingency) of the world. They are exercises in armchair rationalism,
> and as such contingent facts that can only be derived from observation
> cannot be permitted. Having said that, the armchair rationalist is
> allowed to appeal to some very broad facts, such as his own existence,
> and the *appearance* of contingency -- the fact that he sees some
> things in the world around him, and others just are not there. Without
> his own existence, the simplest non-contingent theory of reality is
> that nothing exists! Apparent contingency, the finiteness, of the
> experienced world is a trickier issue.
>
> The Case Against Mathematical Platonism
> The epistemic case for mathematical Platonism is be argued on the basis
> of the objective nature of mathematical truth. Superficially, it seems
> persuasive that objectivity requires objects. However, the basic case
> for the objectivity of mathematics is the tendency of mathematicians to
> agree about the answers to mathematical problems; this can be explained
> by noting that mathematical logic is based on axioms and rules of
> inference, and different mathematicians following the same rules will
> tend to get the same answers , like different computers running the
> same problem.
>
> The semantic case for mathematical Platonism is based on the idea that
> the terms in a mathematical sentence must mean something, and therefore
> must refer to objects. It can be argued on general linguistic grounds
> that not all meaning is reference to some kind of object outside the
> head. Some meaning is sense, some is reference. That establishes the
> possibility that mathematical terms do not have references. What
> establishes it is as likely and not merely possible is the obeservation
> that nothing like empirical investigation is needed to establish the
> truth of mathematical statements. Mathematical truth is arrived at by a
> purely conceptual process, which is what would be expected if
> mathematical meaning were restricted to the Sense, the "in the head"
> component of meaning.
>
> The logical case for mathematical Platonism is based on the idea that
> mathematical statements are true, and make existence claims. That they
> are true is not disputed by the anti-Platonist, who must therefore
> claim that mathematical existence claims are somehow weaker than other
> existence claims -- perhaps merely metaphorical. That the the word
> "exists" means different things in different contexts is easily
> established.
>
> For one thing, this is already conceded by Platonists! Platonists think
> Platonic existence is eternal, immaterial non-spatial, and so on,
> unlike the Earthly existence of material bodies. For another, we are
> already used to contextualising the meaning of "exists". We agree with
> both: "helicopters exist"; and "helicopters don't exist in Middle
> Earth". (People who base their entire anti-Platonic philosophy are on
> this idea are called fictionalists. However, mathematics is not a
> fiction because it is not a free creation. Mathematicians are
> constrained by consistency and non-contradiction in a way that authors
> are not. Dr Watson's fictional existence is intact despite the fact
> that he is sometimes called John and sometimes James in Conan Doyle's
> stories).
>
> A possible counter argument by the Platonist is that the downgrading of
> mathematical existence to a mere metaphor is arbitrary. The
> anti-Platonist must show that a consistent standard is being applied.
> This it is possible to do; the standard is to take the meaning of
> existence in the context of a particular proposition to relate to the
> means of justification of the proposition. Since ordinary statements
> are confirmed empirically, "exists" means "can be perceived" in that
> context. Since sufficient grounds for asserting the existence of
> mathematical objects are that it is does not contradict anything else
> in mathematics, mathematical existence just amounts to concpetual
> non-contradictoriness.
>
> (Incidentally, this approach answers a question about mathematical and
> empirical truth. The anti-Platonists wants the two kinds of truth to be
> different, but also needs them to be related so as to avoid the charge
> that one class of statement is not true at all. This can be achieved
> because empirical statements rest on non-contradiction in order to
> achive correspondence. If an empricial observation fails co correspond
> to a statemet, there is a contradiction between them. Thus
> non-contradiciton is a necessary but insufficient justification for
> truth in empircal statements, but a sufficient one for mathematical
> statements).
>
> The forgoing establishes that Platonism is dispensible. What
> establishes that anti-Platonism is actually preferable? One point in
> its favour is that it is simpler.
>
> All the facts about mathematical truth and methodology can be
> established without appeal to the actual existence of mathematical
> objects. In fact, the lack of such objects actually explains the
> objectivity and necessity of maths. Mathematical statements are
> necessarily true because there are no possible circumstances that make
> them false; there are no possible circumstances that would make them
> false because they do not refer to anything external. This is much
> simpler than the Platonist alternative that mathematical statements:
> 1) have referents
> which are
> 2) unchanging and eternal, unlike anything anyone has actually seen
> and thereby
> 3) explain the necessity (invariance) of mathematical statements
> without
> 4) performing any other role -- they are not involved inbr />
> mathematical proof.
>
> For and Against Matter
> Matter is a bare substrate with no properties of its own. The question
> may well be asked at this point: what roles does it perform ? Why not
> dispense with matter and just have bundles of properties -- what does
> matter add to a merely abstract set of properties? The answer is that
> not all bundles of posible properties are instantiated, that they
> exist.
>
> What does it mean to say something exists ? "..exists" is a meaningful
> predicate of concepts rather than things. The thing must exist in some
> sense to be talked about. But if it existed full, a statement like
> "Nessie doesn't exist" would be a contradiction ...it would amount to
> "the existing thing Nessie doesnt exist". However, if we take that the
> "some sense" in which the subject of an "...exists" predicate exists is
> only initially as a concept, we can then say whether or not the concept
> has something to refer to. Thus "Bigfoot exists" would mean "the
> concept 'Bigfoot' has a referent".
>
> What matter adds to a bundle of properties is existence. A non-existent
> bundle of properties is a mere concept, a mere possibility. Thus the
> concept of matter is very much tied to the idea of contingency or
> "somethingism" -- the idea that only certain possible things exist.
>
> The other issue matter is able to explain as a result of having no
> properties of its own is the issue of change and time. For change to be
> distinguishable from mere succession, it must be change in something.
> It could be a contingent natural law that certain properties never
> change. However, with a propertiless substrate, it becomes a logical
> necessity that the substrate endures through change; since all changes
> are changes in properties, a propertiless substrate cannot itself
> change and must endure through change. In more detail here
>
> The Case Against Mathematial Monism
>
> Mathematical monism is both too broad and too narrow.
>
> Too broad: If I am just a mathematical structure, I should have a much
> wider range of experience than I do. There is a mathemtical structure
> corresponding to myself with all my experiences up to time T. There is
> a vast array of mathematical structures corresponding to other versions
> of me with having a huge range of experiences -- ordinary ones, like
> continuing to type, extraordinary ones like seeing my computer sudenly
> turn into bowl of petunias. All these versions of me share the memories
> of the "me" who is writing this, so they all identify themselves as me.
> Remember, that for mathematical monism it is only necessary that a
> possible experience has a mathematical description. This is known as
> the White Rabbit problem. If we think in terms of multiverse theories,
> we would say that there is one "me" in this universe and other "me's"
> in other universes,a nd they are kept out of contact with each other.
> The question is whether a purely mathematical scheme has enough
> resources to impose isolation or otherwise remove the White Rabbit
> problem.
>
> Too narrow: there are a number of prima-facie phenomena which a purely
> mathematical approach struggles to deal with.
>
> * space
> * time
> * consciousness
> * causality
> * necessity/contingency
>
> Why space ? It is tempting to think that if a number of, or some other
> mathematical entity, occurs in a set with other numbers, that is, as it
> were, a "space" which is disconnected from other sets, so that a set
> forms a natural model of an *isolated* universe withing a multiverse, a
> universe which does not suffer from the White Rabbit problem. However,
> maths per se does not work that way. The number "2" that appears in the
> set of even numbers is exactly the same number "2" that appears in the
> list of numbers less than 10. It does not acquire any further
> characteristics from its context.
>
> The time issue should be obvious. Mathematics is tradionally held to
> deal with timeless, eternal truths. This is reflected in the metpahor
> of mathematical truth being discovered not found (which, in line with
> my criticism of Platonism, should not be taken to seriously). It could
> be objected that physics can model time mathematically; it can be
> objected right back that it does so by spatialising time, by turning it
> into just another dimension, in which nothing really changes, and
> nothing passes. Some even go so far as to insist that this model is
> what time "really" is, which is surely a case of mistaking the map for
> the territory.
>
> Consciousness is a problem for all forms of materialism and physicalism
> to some extent, but it is possible to discern where the problem is
> particularly acute. There is no great problem with the idea that matter
> considered as a bare substrate can have mental properities. Any
> inability to have mental properties would itself be a property and
> therefore be inconsistent with the bareness of a bare substrate. The
> "subjectivity" of conscious states, often treated as "inherent" boils
> down to a problem of communicating one's qualia -- how one feels, how
> things seem. Thus it is not truly inherent but depends on the means of
> communication being used. Feelings and seemings can be more readily
> communicated in artistic, poetic language, and least readily in
> scientific, technical language. Since the harder, more technical a
> science is, the more mathematical it is, the communication problem is
> at its most acute in a purely mathematical langauge. Thus the problem
> with physicalism is not its posit of matter (as a bare substrate) but
> its other posit, that all properties are physical. Since physics is
> mathematical, that amounts to the claim that all properties are
> mathematical (or at least mathematically describable). In making the
> transition from a physicalist world-view to a mathematical one, the
> concept of a material substrate is abandoned (although it was never a
> problem for consciousness) and mathematical properties become the only
> possible basis for qualia. Qualia have to be reducible to, or
> identifiable with, mathematical properties, if they exist at all. This
> means that the problem for consciousness becomes extreme, since there
> is no longer the possibility of qualia existing in their own right, as
> properties of a material substrate, without supervening on
> mathematically describable properties.
>
> The interesting thing is that these two problems can be used to solve
> each other to some extent. if we allow extra-mathemtical properties
> into our universe, we can use them to solve the White Rabbit problem.
> There are two ways of doing this: We can claim either:-
>
> * White Rabbit universes don't exist at all
> * White Rabbit universes are causally separated from us (or remote
> in space)
>
> The first is basically a reversion to a single-universe theory (1).
> Mathematical monists sometimes complain that they can't see what role
> matter plays. One way of seeing its role is as a solution to the WR
> problem. For the non-Platonist, most mathematical entitites have a
> "merely abstract" existence. Only a subset truly, conceretely, exist.
> There is an extra factor that the priveleged few have. What is it ?
> Materiality. For the physicalist, matter is the token of existence.
> Maerial things, exist, immaterial ones don't.
>
> The second moves on from a Mathematical Multiverse to a physical one
> (3). The interesting thing about the second variety of
> non-just-mathematical monism is that as well as addressing the White
> Rabbit problem, it removes some further contingency. If the matter,
> physical laws, and so on, are logically possible, then the general
> approach of arguing for a universe/multiverse on the grounds of
> removing contingency must embrace them -- otherwise it would be a
> contingent fact that the universe/multiverse consists of nothing but
> mathematics.
>
> Peter D Jones 05/05/2006
>
>
> >
>
>
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Received on Tue Nov 07 2006 - 13:09:30 PST